Cubic critical, medium range

Percentage Accurate: 31.5% → 99.4%

Time: 4.1s

Alternatives: 8

Speedup: 2.9×

Specification

\[\left(\left(1.1102230246251565 \cdot 10^{-16} < a \land a < 9007199254740992\right) \land \left(1.1102230246251565 \cdot 10^{-16} < b \land b < 9007199254740992\right)\right) \land \left(1.1102230246251565 \cdot 10^{-16} < c \land c < 9007199254740992\right)\]

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Initial Program: 31.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))

C

double code(double a, double b, double c) {
	return (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-b + sqrt(((b * b) - ((3.0d0 * a) * c)))) / (3.0d0 * a)
end function

Java

public static double code(double a, double b, double c) {
	return (-b + Math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
}

Python

def code(a, b, c):
	return (-b + math.sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a)

Julia

function code(a, b, c)
	return Float64(Float64(Float64(-b) + sqrt(Float64(Float64(b * b) - Float64(Float64(3.0 * a) * c)))) / Float64(3.0 * a))
end

MATLAB

function tmp = code(a, b, c)
	tmp = (-b + sqrt(((b * b) - ((3.0 * a) * c)))) / (3.0 * a);
end

Wolfram

code[a_, b_, c_] := N[(N[((-b) + N[Sqrt[N[(N[(b * b), $MachinePrecision] - N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / N[(3.0 * a), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{\left(3 \cdot a\right) \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* (* 3.0 a) c) (* (- (- b) (sqrt (fma -3.0 (* c a) (* b b)))) (* a 3.0))))

C

double code(double a, double b, double c) {
	return ((3.0 * a) * c) / ((-b - sqrt(fma(-3.0, (c * a), (b * b)))) * (a * 3.0));
}

Julia

function code(a, b, c)
	return Float64(Float64(Float64(3.0 * a) * c) / Float64(Float64(Float64(-b) - sqrt(fma(-3.0, Float64(c * a), Float64(b * b)))) * Float64(a * 3.0)))
end

Wolfram

code[a_, b_, c_] := N[(N[(N[(3.0 * a), $MachinePrecision] * c), $MachinePrecision] / N[(N[((-b) - N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[(a * 3.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

3. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. associate--r+ N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. lower-*.f64 99.2

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(c \cdot a\right) \cdot -3\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(c \cdot a\right) \cdot -3}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{c \cdot \left(a \cdot -3\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot 3\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   17. lower-neg.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(-c\right)} \cdot \left(\mathsf{neg}\left(a \cdot 3\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\left(-c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   19. *-commutative N/A

       \[\leadsto \frac{\left(-c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   20. distribute-lft-neg-in N/A

       \[\leadsto \frac{\left(-c\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   21. metadata-eval N/A

       \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{-3} \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   22. lift-*.f64 99.4

       \[\leadsto \frac{\left(-c\right) \cdot \color{blue}{\left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(-3 \cdot a\right) \cdot \left(-c\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. lift-neg.f64 N/A

      \[\leadsto \frac{\left(-3 \cdot a\right) \cdot \color{blue}{\left(\mathsf{neg}\left(c\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. distribute-rgt-neg-in N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(-3 \cdot a\right) \cdot c\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. distribute-lft-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(-3 \cdot a\right)\right) \cdot c}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\left(\mathsf{neg}\left(\color{blue}{-3 \cdot a}\right)\right) \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. distribute-lft-neg-in N/A

      \[\leadsto \frac{\color{blue}{\left(\left(\mathsf{neg}\left(-3\right)\right) \cdot a\right)} \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. metadata-eval N/A

      \[\leadsto \frac{\left(\color{blue}{3} \cdot a\right) \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(a \cdot 3\right)} \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(a \cdot 3\right)} \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   11. lower-*.f64 99.4

       \[\leadsto \frac{\color{blue}{\left(a \cdot 3\right) \cdot c}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(a \cdot 3\right)} \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   13. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(3 \cdot a\right)} \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   14. lower-*.f64 99.4

       \[\leadsto \frac{\color{blue}{\left(3 \cdot a\right)} \cdot c}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

9. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
10. Add Preprocessing

Alternative 2: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{3 \cdot \left(a \cdot c\right)}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* 3.0 (* a c)) (* (* (- (+ (sqrt (fma -3.0 (* a c) (* b b))) b)) a) 3.0)))

C

double code(double a, double b, double c) {
	return (3.0 * (a * c)) / ((-(sqrt(fma(-3.0, (a * c), (b * b))) + b) * a) * 3.0);
}

Julia

function code(a, b, c)
	return Float64(Float64(3.0 * Float64(a * c)) / Float64(Float64(Float64(-Float64(sqrt(fma(-3.0, Float64(a * c), Float64(b * b))) + b)) * a) * 3.0))
end

Wolfram

code[a_, b_, c_] := N[(N[(3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[((-N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision]) * a), $MachinePrecision] * 3.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

3. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. associate--r+ N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. lower-*.f64 99.2

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \color{blue}{\left(a \cdot 3\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3}} \]

8. Taylor expanded in a around 0

   \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{3 \cdot \color{blue}{\left(a \cdot c\right)}}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3} \]

   2. lower-*.f64 99.2

      \[\leadsto \frac{3 \cdot \left(a \cdot \color{blue}{c}\right)}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3} \]

10. Applied rewrites 99.2%

    \[\leadsto \frac{\color{blue}{3 \cdot \left(a \cdot c\right)}}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3} \]
11. Add Preprocessing

Alternative 3: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.2 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{a \cdot \mathsf{fma}\left(-6, b, 4.5 \cdot \frac{a \cdot c}{b}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.2e-5)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* a 3.0))
   (/ (* (- c) (* -3.0 a)) (* a (fma -6.0 b (* 4.5 (/ (* a c) b)))))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.2e-5) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (a * 3.0);
	} else {
		tmp = (-c * (-3.0 * a)) / (a * fma(-6.0, b, (4.5 * ((a * c) / b))));
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.2e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(Float64(Float64(-c) * Float64(-3.0 * a)) / Float64(a * fma(-6.0, b, Float64(4.5 * Float64(Float64(a * c) / b)))));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.2e-5], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(N[((-c) * N[(-3.0 * a), $MachinePrecision]), $MachinePrecision] / N[(a * N[(-6.0 * b + N[(4.5 * N[(N[(a * c), $MachinePrecision] / b), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.19999999999999977e-5

   1. Initial program 71.9%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}}{3 \cdot a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

      4. sub-negate1-reverse N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      5. lower--.f64 71.9

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

      7. sub-negate1 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}} - b}{3 \cdot a} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right) + b \cdot b}} - b}{3 \cdot a} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right) \cdot c}\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right)} \cdot c\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b} - b}{3 \cdot a} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), a \cdot c, b \cdot b\right)}} - b}{3 \cdot a} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{-3}, a \cdot c, b \cdot b\right)} - b}{3 \cdot a} \]

      15. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)} - b}{3 \cdot a} \]

      16. lower-*.f64 71.9

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)} - b}{3 \cdot a} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{3 \cdot a}} \]

      18. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{a \cdot 3}} \]

      19. lower-*.f64 71.9

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{a \cdot 3}} \]

   3. Applied rewrites 71.9%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{a \cdot 3}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right) + b \cdot b}} - b}{a \cdot 3} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + -3 \cdot \left(c \cdot a\right)}} - b}{a \cdot 3} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)}} - b}{a \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      5. pow2 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      6. exp-to-pow N/A

         \[\leadsto \frac{\sqrt{\color{blue}{e^{\log b \cdot 2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      7. lift-log.f64 N/A

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b} \cdot 2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b \cdot 2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{e^{\log b \cdot 2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{3} \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - 3 \cdot \color{blue}{\left(c \cdot a\right)}} - b}{a \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - 3 \cdot \color{blue}{\left(a \cdot c\right)}} - b}{a \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{\left(3 \cdot a\right) \cdot c}} - b}{a \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{\left(3 \cdot a\right)} \cdot c} - b}{a \cdot 3} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{\left(3 \cdot a\right) \cdot c}} - b}{a \cdot 3} \]

      16. lift-exp.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{e^{\log b \cdot 2}} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b \cdot 2}} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      18. lift-log.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b} \cdot 2} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      19. exp-to-pow N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      20. pow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(3 \cdot a\right) \cdot c}} - b}{a \cdot 3} \]

      23. *-commutative N/A

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}} - b}{a \cdot 3} \]

      24. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \color{blue}{\left(3 \cdot a\right)}} - b}{a \cdot 3} \]

      25. *-commutative N/A

          \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}} - b}{a \cdot 3} \]

      26. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}} - b}{a \cdot 3} \]

      27. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(c\right)\right) \cdot \left(a \cdot 3\right)}} - b}{a \cdot 3} \]

      28. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(c\right)\right) \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \]

      29. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{\left(\mathsf{neg}\left(c \cdot \left(a \cdot 3\right)\right)\right)}} - b}{a \cdot 3} \]

   5. Applied rewrites 71.9%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}} - b}{a \cdot 3} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot -3}\right)} - b}{a \cdot 3} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(c \cdot a\right)}\right)} - b}{a \cdot 3} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \color{blue}{\left(c \cdot a\right)}\right)} - b}{a \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \color{blue}{\left(a \cdot c\right)}\right)} - b}{a \cdot 3} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right) \cdot c}\right)} - b}{a \cdot 3} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot c\right)} - b}{a \cdot 3} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} \cdot c\right)} - b}{a \cdot 3} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(a \cdot 3\right)\right) \cdot c}\right)} - b}{a \cdot 3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot c\right)} - b}{a \cdot 3} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot a\right) \cdot c\right)} - b}{a \cdot 3} \]

      15. lift-*.f64 71.9

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right)} \cdot c\right)} - b}{a \cdot 3} \]

   7. Applied rewrites 71.9%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right) \cdot c}\right)} - b}{a \cdot 3} \]

   if 4.19999999999999977e-5 < b

   1. Initial program 28.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   3. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      4. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      8. lower-*.f64 99.2

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      2. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      3. +-inverses N/A

         \[\leadsto \frac{\color{blue}{0} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      4. sub0-neg N/A

         \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(c \cdot a\right) \cdot -3\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(c \cdot a\right) \cdot -3}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      6. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{c \cdot \left(a \cdot -3\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{neg}\left(c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{neg}\left(c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      10. distribute-lft-neg-in N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      12. metadata-eval N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      14. *-commutative N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      16. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot 3\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      17. lower-neg.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(-c\right)} \cdot \left(\mathsf{neg}\left(a \cdot 3\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      18. lift-*.f64 N/A

          \[\leadsto \frac{\left(-c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      19. *-commutative N/A

          \[\leadsto \frac{\left(-c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      20. distribute-lft-neg-in N/A

          \[\leadsto \frac{\left(-c\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      21. metadata-eval N/A

          \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{-3} \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      22. lift-*.f64 99.4

          \[\leadsto \frac{\left(-c\right) \cdot \color{blue}{\left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. Applied rewrites 99.4%

      \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. Taylor expanded in a around 0

      \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{\color{blue}{a \cdot \left(-6 \cdot b + \frac{9}{2} \cdot \frac{a \cdot c}{b}\right)}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{a \cdot \color{blue}{\left(-6 \cdot b + \frac{9}{2} \cdot \frac{a \cdot c}{b}\right)}} \]

      2. lower-fma.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{a \cdot \mathsf{fma}\left(-6, \color{blue}{b}, \frac{9}{2} \cdot \frac{a \cdot c}{b}\right)} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{a \cdot \mathsf{fma}\left(-6, b, \frac{9}{2} \cdot \frac{a \cdot c}{b}\right)} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{a \cdot \mathsf{fma}\left(-6, b, \frac{9}{2} \cdot \frac{a \cdot c}{b}\right)} \]

      5. lower-*.f64 91.7

         \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{a \cdot \mathsf{fma}\left(-6, b, 4.5 \cdot \frac{a \cdot c}{b}\right)} \]

   10. Applied rewrites 91.7%

       \[\leadsto \frac{\left(-c\right) \cdot \left(-3 \cdot a\right)}{\color{blue}{a \cdot \mathsf{fma}\left(-6, b, 4.5 \cdot \frac{a \cdot c}{b}\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 99.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{-3 \cdot \left(a \cdot c\right)}{\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right) \cdot \left(3 \cdot a\right)} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (/ (* -3.0 (* a c)) (* (+ (sqrt (fma -3.0 (* a c) (* b b))) b) (* 3.0 a))))

C

double code(double a, double b, double c) {
	return (-3.0 * (a * c)) / ((sqrt(fma(-3.0, (a * c), (b * b))) + b) * (3.0 * a));
}

Julia

function code(a, b, c)
	return Float64(Float64(-3.0 * Float64(a * c)) / Float64(Float64(sqrt(fma(-3.0, Float64(a * c), Float64(b * b))) + b) * Float64(3.0 * a)))
end

Wolfram

code[a_, b_, c_] := N[(N[(-3.0 * N[(a * c), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sqrt[N[(-3.0 * N[(a * c), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] + b), $MachinePrecision] * N[(3.0 * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

3. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. associate--r+ N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. lower-*.f64 99.2

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

7. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{-3 \cdot \left(a \cdot c\right)}{\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right) \cdot \left(3 \cdot a\right)}} \]
8. Add Preprocessing

Alternative 5: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(b, b, \left(-3 \cdot a\right) \cdot c\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.5e-5)
   (/ (- (sqrt (fma b b (* (* -3.0 a) c))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.5e-5) {
		tmp = (sqrt(fma(b, b, ((-3.0 * a) * c))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.5e-5)
		tmp = Float64(Float64(sqrt(fma(b, b, Float64(Float64(-3.0 * a) * c))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.5e-5], N[(N[(N[Sqrt[N[(b * b + N[(N[(-3.0 * a), $MachinePrecision] * c), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.50000000000000028e-5

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}}{3 \cdot a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

      4. sub-negate1-reverse N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      5. lower--.f64 72.0

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

      7. sub-negate1 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}} - b}{3 \cdot a} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right) + b \cdot b}} - b}{3 \cdot a} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right) \cdot c}\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right)} \cdot c\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b} - b}{3 \cdot a} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), a \cdot c, b \cdot b\right)}} - b}{3 \cdot a} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{-3}, a \cdot c, b \cdot b\right)} - b}{3 \cdot a} \]

      15. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)} - b}{3 \cdot a} \]

      16. lower-*.f64 72.0

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)} - b}{3 \cdot a} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{3 \cdot a}} \]

      18. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{a \cdot 3}} \]

      19. lower-*.f64 72.0

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{a \cdot 3}} \]

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{a \cdot 3}} \]
   4. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{-3 \cdot \left(c \cdot a\right) + b \cdot b}} - b}{a \cdot 3} \]

      2. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + -3 \cdot \left(c \cdot a\right)}} - b}{a \cdot 3} \]

      3. fp-cancel-sign-sub-inv N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)}} - b}{a \cdot 3} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      5. pow2 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      6. exp-to-pow N/A

         \[\leadsto \frac{\sqrt{\color{blue}{e^{\log b \cdot 2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      7. lift-log.f64 N/A

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b} \cdot 2} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b \cdot 2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      9. lift-exp.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{e^{\log b \cdot 2}} - \left(\mathsf{neg}\left(-3\right)\right) \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      10. metadata-eval N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{3} \cdot \left(c \cdot a\right)} - b}{a \cdot 3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - 3 \cdot \color{blue}{\left(c \cdot a\right)}} - b}{a \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - 3 \cdot \color{blue}{\left(a \cdot c\right)}} - b}{a \cdot 3} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{\left(3 \cdot a\right) \cdot c}} - b}{a \cdot 3} \]

      14. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{\left(3 \cdot a\right)} \cdot c} - b}{a \cdot 3} \]

      15. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\log b \cdot 2} - \color{blue}{\left(3 \cdot a\right) \cdot c}} - b}{a \cdot 3} \]

      16. lift-exp.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{e^{\log b \cdot 2}} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b \cdot 2}} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      18. lift-log.f64 N/A

          \[\leadsto \frac{\sqrt{e^{\color{blue}{\log b} \cdot 2} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      19. exp-to-pow N/A

          \[\leadsto \frac{\sqrt{\color{blue}{{b}^{2}} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      20. pow2 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      21. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} - \left(3 \cdot a\right) \cdot c} - b}{a \cdot 3} \]

      22. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{\left(3 \cdot a\right) \cdot c}} - b}{a \cdot 3} \]

      23. *-commutative N/A

          \[\leadsto \frac{\sqrt{b \cdot b - \color{blue}{c \cdot \left(3 \cdot a\right)}} - b}{a \cdot 3} \]

      24. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \color{blue}{\left(3 \cdot a\right)}} - b}{a \cdot 3} \]

      25. *-commutative N/A

          \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}} - b}{a \cdot 3} \]

      26. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{b \cdot b - c \cdot \color{blue}{\left(a \cdot 3\right)}} - b}{a \cdot 3} \]

      27. fp-cancel-sub-sign-inv N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(c\right)\right) \cdot \left(a \cdot 3\right)}} - b}{a \cdot 3} \]

      28. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b} + \left(\mathsf{neg}\left(c\right)\right) \cdot \left(a \cdot 3\right)} - b}{a \cdot 3} \]

      29. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{b \cdot b + \color{blue}{\left(\mathsf{neg}\left(c \cdot \left(a \cdot 3\right)\right)\right)}} - b}{a \cdot 3} \]

   5. Applied rewrites 72.0%

      \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(b, b, \left(c \cdot a\right) \cdot -3\right)}} - b}{a \cdot 3} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(c \cdot a\right) \cdot -3}\right)} - b}{a \cdot 3} \]

      2. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{-3 \cdot \left(c \cdot a\right)}\right)} - b}{a \cdot 3} \]

      3. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \color{blue}{\left(c \cdot a\right)}\right)} - b}{a \cdot 3} \]

      4. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, -3 \cdot \color{blue}{\left(a \cdot c\right)}\right)} - b}{a \cdot 3} \]

      5. associate-*r* N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right) \cdot c}\right)} - b}{a \cdot 3} \]

      6. metadata-eval N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right) \cdot c\right)} - b}{a \cdot 3} \]

      7. distribute-lft-neg-in N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)} \cdot c\right)} - b}{a \cdot 3} \]

      8. *-commutative N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\mathsf{neg}\left(a \cdot 3\right)\right) \cdot c}\right)} - b}{a \cdot 3} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      12. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right) \cdot c\right)} - b}{a \cdot 3} \]

      13. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)} \cdot c\right)} - b}{a \cdot 3} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \left(\color{blue}{-3} \cdot a\right) \cdot c\right)} - b}{a \cdot 3} \]

      15. lift-*.f64 72.0

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right)} \cdot c\right)} - b}{a \cdot 3} \]

   7. Applied rewrites 72.0%

      \[\leadsto \frac{\sqrt{\mathsf{fma}\left(b, b, \color{blue}{\left(-3 \cdot a\right) \cdot c}\right)} - b}{a \cdot 3} \]

   if 4.50000000000000028e-5 < b

   1. Initial program 28.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   3. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      4. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      8. lower-*.f64 99.2

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \color{blue}{\left(a \cdot 3\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 83.3

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b}} \]

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 82.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;b \leq 4.5 \cdot 10^{-5}:\\ \;\;\;\;\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{a \cdot 3}\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \frac{c}{b}\\ \end{array} \end{array} \]

FPCore

(FPCore (a b c)
 :precision binary64
 (if (<= b 4.5e-5)
   (/ (- (sqrt (fma -3.0 (* c a) (* b b))) b) (* a 3.0))
   (* -0.5 (/ c b))))

C

double code(double a, double b, double c) {
	double tmp;
	if (b <= 4.5e-5) {
		tmp = (sqrt(fma(-3.0, (c * a), (b * b))) - b) / (a * 3.0);
	} else {
		tmp = -0.5 * (c / b);
	}
	return tmp;
}

Julia

function code(a, b, c)
	tmp = 0.0
	if (b <= 4.5e-5)
		tmp = Float64(Float64(sqrt(fma(-3.0, Float64(c * a), Float64(b * b))) - b) / Float64(a * 3.0));
	else
		tmp = Float64(-0.5 * Float64(c / b));
	end
	return tmp
end

Wolfram

code[a_, b_, c_] := If[LessEqual[b, 4.5e-5], N[(N[(N[Sqrt[N[(-3.0 * N[(c * a), $MachinePrecision] + N[(b * b), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] - b), $MachinePrecision] / N[(a * 3.0), $MachinePrecision]), $MachinePrecision], N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if b < 4.50000000000000028e-5

   1. Initial program 72.0%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      2. +-commutative N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \left(-b\right)}}{3 \cdot a} \]

      3. lift-neg.f64 N/A

         \[\leadsto \frac{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} + \color{blue}{\left(\mathsf{neg}\left(b\right)\right)}}{3 \cdot a} \]

      4. sub-negate1-reverse N/A

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      5. lower--.f64 72.0

         \[\leadsto \frac{\color{blue}{\sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} - b}}{3 \cdot a} \]

      6. lift--.f64 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b - \left(3 \cdot a\right) \cdot c}} - b}{3 \cdot a} \]

      7. sub-negate1 N/A

         \[\leadsto \frac{\sqrt{\color{blue}{b \cdot b + \left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right)}} - b}{3 \cdot a} \]

      8. +-commutative N/A

         \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(\left(3 \cdot a\right) \cdot c\right)\right) + b \cdot b}} - b}{3 \cdot a} \]

      9. lift-*.f64 N/A

         \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right) \cdot c}\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      10. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{\left(3 \cdot a\right)} \cdot c\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      11. associate-*l* N/A

          \[\leadsto \frac{\sqrt{\left(\mathsf{neg}\left(\color{blue}{3 \cdot \left(a \cdot c\right)}\right)\right) + b \cdot b} - b}{3 \cdot a} \]

      12. distribute-lft-neg-in N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\left(\mathsf{neg}\left(3\right)\right) \cdot \left(a \cdot c\right)} + b \cdot b} - b}{3 \cdot a} \]

      13. lower-fma.f64 N/A

          \[\leadsto \frac{\sqrt{\color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(3\right), a \cdot c, b \cdot b\right)}} - b}{3 \cdot a} \]

      14. metadata-eval N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(\color{blue}{-3}, a \cdot c, b \cdot b\right)} - b}{3 \cdot a} \]

      15. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)} - b}{3 \cdot a} \]

      16. lower-*.f64 72.0

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, \color{blue}{c \cdot a}, b \cdot b\right)} - b}{3 \cdot a} \]

      17. lift-*.f64 N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{3 \cdot a}} \]

      18. *-commutative N/A

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{a \cdot 3}} \]

      19. lower-*.f64 72.0

          \[\leadsto \frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{\color{blue}{a \cdot 3}} \]

   3. Applied rewrites 72.0%

      \[\leadsto \color{blue}{\frac{\sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)} - b}{a \cdot 3}} \]

   if 4.50000000000000028e-5 < b

   1. Initial program 28.7%

      \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

      3. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

      4. associate-/l/ N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   3. Applied rewrites 29.6%

      \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
   4. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      2. lift-fma.f64 N/A

         \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      3. +-commutative N/A

         \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      4. associate--r+ N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      6. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      7. *-commutative N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

      8. lower-*.f64 99.2

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. Applied rewrites 99.2%

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \color{blue}{\left(a \cdot 3\right)}} \]

      3. associate-*r* N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

      4. lower-*.f64 N/A

         \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

   7. Applied rewrites 99.2%

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3}} \]

   8. Taylor expanded in a around 0

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
   9. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b}} \]

      2. lower-/.f64 83.3

         \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b}} \]

   10. Applied rewrites 83.3%

       \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 81.2% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ -0.5 \cdot \frac{c}{b} \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 (* -0.5 (/ c b)))

C

double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (-0.5d0) * (c / b)
end function

Java

public static double code(double a, double b, double c) {
	return -0.5 * (c / b);
}

Python

def code(a, b, c):
	return -0.5 * (c / b)

Julia

function code(a, b, c)
	return Float64(-0.5 * Float64(c / b))
end

MATLAB

function tmp = code(a, b, c)
	tmp = -0.5 * (c / b);
end

Wolfram

code[a_, b_, c_] := N[(-0.5 * N[(c / b), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 31.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

3. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. associate--r+ N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. lower-*.f64 99.2

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
6. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]

   2. lift-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \color{blue}{\left(a \cdot 3\right)}} \]

   3. associate-*r* N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

   4. lower-*.f64 N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot a\right) \cdot 3}} \]

7. Applied rewrites 99.2%

   \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}{\color{blue}{\left(\left(-\left(\sqrt{\mathsf{fma}\left(-3, a \cdot c, b \cdot b\right)} + b\right)\right) \cdot a\right) \cdot 3}} \]

8. Taylor expanded in a around 0

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{c}{b}} \]
9. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \frac{-1}{2} \cdot \color{blue}{\frac{c}{b}} \]

   2. lower-/.f64 81.2

      \[\leadsto -0.5 \cdot \frac{c}{\color{blue}{b}} \]

10. Applied rewrites 81.2%

    \[\leadsto \color{blue}{-0.5 \cdot \frac{c}{b}} \]
11. Add Preprocessing

Alternative 8: 3.2% accurate, 50.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b c) :precision binary64 0.0)

C

double code(double a, double b, double c) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(a, b, c)
use fmin_fmax_functions
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = 0.0d0
end function

Java

public static double code(double a, double b, double c) {
	return 0.0;
}

Python

def code(a, b, c):
	return 0.0

Julia

function code(a, b, c)
	return 0.0
end

MATLAB

function tmp = code(a, b, c)
	tmp = 0.0;
end

Wolfram

code[a_, b_, c_] := 0.0

Derivation

1. Initial program 31.5%

   \[\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{3 \cdot a}} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(-b\right) + \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}{3 \cdot a} \]

   3. flip-+ N/A

      \[\leadsto \frac{\color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}}}{3 \cdot a} \]

   4. associate-/l/ N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

   5. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(-b\right) \cdot \left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c} \cdot \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}}{\left(\left(-b\right) - \sqrt{b \cdot b - \left(3 \cdot a\right) \cdot c}\right) \cdot \left(3 \cdot a\right)}} \]

3. Applied rewrites 32.4%

   \[\leadsto \color{blue}{\frac{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)}} \]
4. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{b \cdot b - \mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift-fma.f64 N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(-3 \cdot \left(c \cdot a\right) + b \cdot b\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-commutative N/A

      \[\leadsto \frac{b \cdot b - \color{blue}{\left(b \cdot b + -3 \cdot \left(c \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. associate--r+ N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - -3 \cdot \left(c \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - -3 \cdot \left(c \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. *-commutative N/A

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. lower-*.f64 99.2

      \[\leadsto \frac{\left(b \cdot b - b \cdot b\right) - \color{blue}{\left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

5. Applied rewrites 99.2%

   \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right) - \left(c \cdot a\right) \cdot -3}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   2. lift--.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(b \cdot b - b \cdot b\right)} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   3. +-inverses N/A

      \[\leadsto \frac{\color{blue}{0} - \left(c \cdot a\right) \cdot -3}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   4. sub0-neg N/A

      \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(c \cdot a\right) \cdot -3\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(c \cdot a\right) \cdot -3}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   6. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(c \cdot a\right)} \cdot -3\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   7. associate-*l* N/A

      \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{c \cdot \left(a \cdot -3\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   8. *-commutative N/A

      \[\leadsto \frac{\mathsf{neg}\left(c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \frac{\mathsf{neg}\left(c \cdot \color{blue}{\left(-3 \cdot a\right)}\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   10. distribute-lft-neg-in N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   12. metadata-eval N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\color{blue}{\left(\mathsf{neg}\left(3\right)\right)} \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   13. distribute-lft-neg-in N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \color{blue}{\left(\mathsf{neg}\left(3 \cdot a\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   14. *-commutative N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   15. lift-*.f64 N/A

       \[\leadsto \frac{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   16. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(\mathsf{neg}\left(c\right)\right) \cdot \left(\mathsf{neg}\left(a \cdot 3\right)\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   17. lower-neg.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(-c\right)} \cdot \left(\mathsf{neg}\left(a \cdot 3\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{\left(-c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{a \cdot 3}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   19. *-commutative N/A

       \[\leadsto \frac{\left(-c\right) \cdot \left(\mathsf{neg}\left(\color{blue}{3 \cdot a}\right)\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   20. distribute-lft-neg-in N/A

       \[\leadsto \frac{\left(-c\right) \cdot \color{blue}{\left(\left(\mathsf{neg}\left(3\right)\right) \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   21. metadata-eval N/A

       \[\leadsto \frac{\left(-c\right) \cdot \left(\color{blue}{-3} \cdot a\right)}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

   22. lift-*.f64 99.4

       \[\leadsto \frac{\left(-c\right) \cdot \color{blue}{\left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

7. Applied rewrites 99.4%

   \[\leadsto \frac{\color{blue}{\left(-c\right) \cdot \left(-3 \cdot a\right)}}{\left(\left(-b\right) - \sqrt{\mathsf{fma}\left(-3, c \cdot a, b \cdot b\right)}\right) \cdot \left(a \cdot 3\right)} \]

9. Applied rewrites 3.2%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025111 
(FPCore (a b c)
  :name "Cubic critical, medium range"
  :precision binary64
  :pre (and (and (and (< 1.1102230246251565e-16 a) (< a 9007199254740992.0)) (and (< 1.1102230246251565e-16 b) (< b 9007199254740992.0))) (and (< 1.1102230246251565e-16 c) (< c 9007199254740992.0)))
  (/ (+ (- b) (sqrt (- (* b b) (* (* 3.0 a) c)))) (* 3.0 a)))